Within the nuclear shell model, we investigate the correction $delta_{RO}$ to the Fermi matrix element due to a mismatch between proton and neutron single-particle radial wave functions. Eight superallowed $0^+ to 0^+$ $beta$ decays in the $sd$-shell, comprised of $^{22}$Mg, $^{26m}$Al, $^{26}$Si, $^{30}$S, $^{34}$Cl, $^{34}$Ar, $^{38m}$K and $^{38}$Ca are re-examined. The radial wave functions are obtained from a spherical Woods-Saxon potential whose parametrizations are optimized in a consistent adjustment of the depth and the length parameter to relevant experimental observables, such as nucleon separation energies and charge radii, respectively. The chosen fit strategy eliminates the strong dependence of the radial mismatch correction to a specific parametrization, except for calculations with an additional surface-peaked term. As an improvement, our model proposes a new way to calculate the charge radii, based on a parentage expansion which accounts for correlations beyond the extreme independent-particle model. Apart from the calculations with a surface-peak term and the cases where we used a different model space, the new sets of $delta_{RO}$ are in general agreement with the earlier result of Towner and Hardy [1]. Small differences of the corrected average $overline{mathcal{F}t}$ value are observed.
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